PROBABILITY 779 whence n (35). 1 Thus, if the a priori probability of the disease ill all the patients was xV, and 3 out of 4 have the disease where A is observed, and also 3 out of 4 where B is observed, the chance is that the disease exists when both symptoms are present. This question illustrates the exceeding delicacy and care required in reasoning on probabilities. If we had combined the two given probabilities in the usual way without considering the a priori value (as would be correct if this were quite unknown, or = ^) we should have had n = + (1 ^f}(l TZT ) The fallacy of so doing will appear if we consider a large population, and a very uncommon disease, and that the latter is observed to exist in 4 the cases where the symptom A occurs, and also in 4 for the symptom B ; this formula would give ^ for the chance when both are" present. This is clearly absurd ; for, both the disease and the symptoms being by hypothesis extremely rare, and the symptoms being independent, that is, having no connexion with each other, it is next to impossible that any one individual of the ^N(A) calling N(A) the number who have the symptom A who have not the disease should also be comprised in the ^N(B) who have not the disease, because this iN(A), ^N(B) are very small numbers (relatively) taken indiscriminately from the whole population who are free from the disease. It is different for the ^-N(A), N(B) cases who have the disease ; these cases all come out of the very small number N(D) who have the disease ; therefore several individuals will be probably common to both ; hence, if both symptoms coexist, it is highly probable that the case is one of the disease. We find from (35) the true probability to be in the present case so that, if only 1 in 1000 have the disease, the chance is 999 to 1, instead of an even one. 39. If a coin thrown m times has turned up head every time, the chance derived from this experience alone that the real facility for head exceeds ^ is, by formula (14), , 7 x ax
I m , 2 + 1 x ax But there is here a very strong a priori presumption that the facility is ^ ; suppose then that there is a very small a priori probability (p) that either in the coin itself or the way it is thrown there is something more favourable to head than to tail ; after the new evidence the probability of this will be Thus if there is an a priori probability TT ftn7, an( l if the coin has turned up head 5 times and never tail, the probability that the facility for head exceeds that for tail becomes 63 60 62 + 1000 1000 40. From art. 19 we see that if a large number of trials m + n be made as to any event, m being favourable, it may be considered certain that the real facility differs from m/(m + n) by a very small fraction at most. If then our a priori idea as to the facility gives it outside the limits derived from formula (21), the evidence from experience will overrule onr a priori presumption. Thus, if a shilling thrown up 1000 times gives head 560 times and tail 440, the evidence thus afforded that the throws were not fair is so much stronger than any antecedent conviction we could have to the contrary that we may conclude with certainty that, from some cause or other, head is more likely than tail. 41. Closely allied to the subject of our present section are the applications of the theory of probabilities to the verdicts of juries, the decisions of courts, and the results of elections. Our limits, 1 Or thus : let N = whole population and n,n the numbers who show the symptoms A and 15 respectively, all these numbers being large. Now aN = whole number who have the disease; -stii, zv n the numbers out of n,n who have it. Now afn, -a ri are both comprised in aX ; and, out of zj n , the number also included in ism is the same fraction of arn that -as n is of aX; that is, the nu nber who have both symptoms and the disease is tf n an =:; ; aN and those who have both symptoms and have not the disease is so that, if both symptoms arc present, the odds that it is a case of the disease are as however, will hardly allow of even a sketch of the methods given by Coudorcet, Laplace, and 1 oisson, as it is not possible to render them intelligible within a short compass. AVe must therefore refer the reader to Todhuntcr s History, as well as the original works of these writers, especially to Poisson s licchcrches sur la Prolubilite des Jugements. 42. We will consider here one remarkable question given by Laplace, because the mathematical difficulty may be solved in a simpler way than by deducing it as a case of a general problem given in his chap, ii., or than Todhunter s method (see his p. 545), which depends on Lejeune Dirichlet s theorem in multiple integrals. An event (suppose the death of a certain person) must have pro ceeded from one ofrc, causes A, B, C, &c., and a tribunal has to pronounce on which is the most probable. Let each member of the tribunal arrange the causes in the order of their probability according to his judgment, after weighing tlie evidence. To compare the presumption thus afforded by any one judge in favour of a specified cause with that afforded by the other judges, we must assign a value to the probability of the cause derived solely from its being, say, the rth on his list. As he is supposed to be unable to pronounce any closer to the truth than to say (suppose) H is more likely than D, D more likely than L, &c., the probability of any cause will be the average value of all those which that probability can have, given simply that it always occupies the same place on the list of the probabilities arranged in order of magnitude. As the sum of the n probabilities is always 1, the question reduces to this Any whole (such as. the number 1) is divided at random into n parts, and the parts are arranged in the order of their magnitude least, second, third, . . . greatest ; this is repeated for the same whole a great number of times ; required the mean value of the least, of the second, &c., parts, up to that of the greatest. A Let the Avhole in question be represented by aline AB = a, and let it be divided at random into n parts by taking n - 1 points indis criminately on it. Let the required mean values be A 3 a, , 2 n, A 3 .... A n , where X 1 , 2 ,A 3 . . . must be constant fractions. As a great number of positions is taken in AB for each of the n points, we may take a as representing that number ; and the whole number N of cases will be N-a"- 1 . The sum of the least parts, in every case, will be S 1 = NA 1 = A. 1 ". Let a small increment, B6= 8, be added on to the line AB at the end B ; the increase in this sum is SS^nA^ -^Sa. But, in dividing the new line A.I, either the n - 1 points all fall on AB as before, or n- 2 fall on AB and 1 on B& (the cases where 2 or more fall on B& are so few we may neglect them). If all fall on AB, the least part is always the same as before except when it is the last, at the end B of the line, and then it is greater than before by Sa ; as it falls last in n -1 of the whole number of trials, the increase in Sj is n - l a n - : 5a. But if one point of division falls on Eb, the number of new cases introduced is (n-l)a n ~-$a ; but, the least part being now an infinitesimal, the sum Sj is not affected ; we have therefore .. To find A 2 , reasoning exactly in the same way, we find that where one point falls on B& and n-2 on AB, as the least part is infinitesimal, the second least part is the least of the n-l parts made by the n- 2 points ; consequently, if we put A J for the value of A! when there are n - 1 parts only, instead of n, . ?^ 2 = ~ 1 + (n - l); but A : = (-!) ~ . . n.-, = n 1 + (n- I)" 1 . In the same way we can show generally that 7lA r =?l" 1 + (n- l) r -l , and thus the required mean value of the rth part is A r = a?i- 1 {- 1 + (-l)- 1 + (-2)- 1 + . . . (n-r+l)- Thus each judge implicitly assigns the probabilities . (36). n-lj n
n-l n
to the causes as they stand on his list, beginning from the lowest. Laplace now says we should add the numbers thus found on the different lists for the cause A, also for B, &c. ; and that cause which has the greatest sum is the most probable. This doubtless seemed
self-evident to him, but ordinary minds will hardly be convinced
